. Differential and integral calculus, an introductory course for colleges and engineering schools. 6. The Cardioid. This is the epicycloidin which b = a. Write down its equations,determine y, and find maximum andminimum abscissas and ordinates. Findthe slope of the tangent at the cusp,and at the points where the curve cutsOY. Calculate y. Draw the curve. 7. Discuss completely the hypocycloid 8. Discusswhen b = 2 a. completely the epicycloid 96. The Involute of the Circle. When in the epicycloid b = qo ,the rolling circle becomes a straight line, and may be regarded asthe taut portion of a stri
. Differential and integral calculus, an introductory course for colleges and engineering schools. 6. The Cardioid. This is the epicycloidin which b = a. Write down its equations,determine y, and find maximum andminimum abscissas and ordinates. Findthe slope of the tangent at the cusp,and at the points where the curve cutsOY. Calculate y. Draw the curve. 7. Discuss completely the hypocycloid 8. Discusswhen b = 2 a. completely the epicycloid 96. The Involute of the Circle. When in the epicycloid b = qo ,the rolling circle becomes a straight line, and may be regarded asthe taut portion of a string woundround the fixed circle and carryinga pencil, P, which traces the curveas the string is wound off the special epicycloid is termed theinvolute of the circle. We obtain itsparametric equations by determin-ing the limiting forms of equations(a), Art. 94, when b = oo. To thisend we write equations (a) in theform. x = a cos 6 + b (cos 6 — cos a + b •) = a cos e+26sin^+1)9sing, y = a sin 6 + Msin 6 — sin —7— g) x = a cos 6 + ad sin (^-=- + 1J = asin0— 2 b cosThese may be written sin ad26 aS_26 aQ_26 §96 • CYCLOIDAL CURVES 133 . ad cm y = a sin 6 — a^cosf^r + 1J0 Sm2 6 ad26 Now, when b = oo, . ad sin l^-r + 1J0 = sin 6, cos (~-r + 1 )6 = cos 0, and — = 1 26Hence the parametric equations of the involute of the circle are x = a(cos 6 + 6 sin 6),) , n y = a(sin0 — 0cos 6). CHAPTER XIII CURVES GIVEN BY POLAR EQUATIONS 97. The Tangent in Polar Coordinates. Let p, 0 be the polar coordinates of the curve point P,and let be the angle between theradius vector and the tangent at seek to express tan in termsof p and 0. Let the coordinates ofQ be p + Ap and 0 + A0. Let PBbe drawn perpendicular to OQ. Then tan 0 = 757:, P£ = p sin A0,andBQ = OQ - OB = p + Ap - p cos A0.
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectcalculu, bookyear1912
